6 fascinating appearances of the Fibonacci numbers in nature

The Italian mathematician Leonardo Fibonacci is widely regarded as the greatest European mathematician of the middle ages. Did you know that Fibonacci introduced the Hindu-Arabic numerals (the decimal system) to the Western world? This system replaced the inefficient and clumsy Roman numerals.

1202 was the year Fibonacci published Liber abaci, a book that revolutionized everything from economics and commerce to education and science. It is quite impossible to imagine how some of the great turning points in Western history could have been achieved without Fibonacci’s Book of Calculation.

In Liber Abaci, the Italian mathematician also introduced a very interesting sequence of numbers with intriguing characteristics. I highly recommend reading The Mathematical Magic of the Fibonacci Numbers, an award-winning collection of information written by Dr. Ron Knott, to understand the remarkable properties of this sequence.

As you may recall from math class, the Fibonacci series can go on infinitely, as it begins with 0, 1, 1 and each subsequent number is created by adding the previous two together: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 … Fn = Fn-1 + Fn-2; with F0 = 0, F1=1

By itself, Fibonacci’s sequence is not very impressive. Still, as you will notice, no other numbers are as ubiquitous as these ones. They have countless applications in modern mathematics and computer science, but what is really fascinating is their discreet but striking appearance in nature.

This series of fabulous numbers is also the mathematical first cousin of φ (Phi), the golden ratio – another special number that has fascinated the human culture for millennia. The golden ratio is the limit of the ratios of consecutive Fibonacci numbers.

φ = 1.6180339887498948482045868343656

If a Fibonacci number is divided by the one before it, the quotient approximates Phi; e.g. 4181/2584 ≈ 1.6180340557275541795665634674923…

The golden ratio is supposed to be the ideal proportion, the most aesthetically pleasing shape to the human eye. What makes the Fibonacci sequence so special? Nature seems to have built itself around it.

6. The ancestral tree of a male honeybee

The ancestral tree of a male honeybee follows Fibonacci’s sequence. Female honeybees (either workers or queens) hatch from an egg that has been fertilized by a male honeybee, but male honeybees are produced by the queen’s unfertilised eggs, so they have a mother but no father. As the above picture clearly shows, the sequence that gives us the number of honeybees in each generation of the ancestral tree of a male bee is 1, 1, 2, 3, 5, 8, 13…

1st generation……..1 offspring

2nd generation…….1 parent

3rd generation…….2 grandparents

4th generation…….3 great-grandparents

5th generation…….5 great-great-grandparents…

The pattern that emerged was reduced to a mathematical model that can be used to find the number of bees in any generation back from the very first honeybee.

5. Leaf and flower arrangements

“Based on a survey of the literature encompassing 650 species and 12,500 specimens, R. Jean estimated that, among plants displaying spiral or multijugate phyllotaxis, about 92 percent of them have Fibonacci phyllotaxis (from the Greek: Phyllo means leaf, and taxis means arrangement)”,

wrote Alfred S. Posamentier and Ingmar Lehmann in The Fabulous Fibonacci Numbers.

In most Aroids, a vast group of beautiful ornamental plants, flowers are arranged in a mathematical series. Clear spirals are visible and the numbers of these spirals are usually a pair of Fibonacci numbers. For example, all the spadices of Anthurium macrolobium present floral spirals matching the Fibonacci numbers eight and five. The Indian Statistical Institute and the Royal Agri-Horticultural Society dedicated an entire study to this topic. It provides solid evidence to support these claims. You can find it online under the name Fibonacci System in Aroids.

Let’s try something. Locate the lowest leaf of a green plant that hasn’t been pruned. Count both the number of times you circle the stem of the plant before arriving at the leaf located directly above the first one (pointing in the same direction), as well as the number of leaves above the lowest located leaf. The number of rotations, of turns in each direction and the number of leaves met will be Fibonacci numbers! Of course, leaf arrangements vary from species to species, but they should all be Fibonacci numbers.

If the number of turns is x and the number of leaves is y, specialists commonly call the leaf arrangement x/y phyllotaxis or x/y spiral. The following ratios are the phyllotaxis ratios of different plants:

1/2 phyllotaxis: elm, basswood, lime, some grasses etc.

1/3 phyllotaxis: beech, hazel, blackberry etc.

2/5 phyllotaxis: oak, apricot, cherry, apple, some roses etc.

3/8 phyllotaxis: cabbages, poplar, pear, hawkweed etc.

5/13 phyllotaxis: willow, almond etc.

13/34 phyllotaxis: some pine trees

4. Pine cones – scale arrangement

Pine cones are good at Fibonacci numbers! There are many species of pine, about 115, all with different characteristics, but what most of them have in common are the Fibonacci spirals of the cones. If you look closely at a pine cone, you will see two distinct spirals. The number of spirals going in each direction typically corresponds to successive Fibonacci numbers. The spirals of the cone from the above picture correspond to 8 and 13. When this occurs, the angle between consecutive scales gradually approximates the golden angle – which is about 137.5 degrees. This angle is responsible for the optimal packing of scales.

Find out more by reading Fibonacci Statistics in Conifers, a very interesting study published by Alfred Brousseau.

3. Fruits

Some fruits like the pineapple, banana, Sharon fruit, apple and more exhibit patterns following the Fibonacci sequence.

Expecting all pineapples to display the Fibonacci sequence is too much to ask, but beautifully grown pineapples with spirally arranged fruitlets still follow a simple rule that illustrates perfection in nature, the Fibonacci sequence.

The pineapple’s hexagonal bracts form three distinct sets of spirals. If we count the number of rows formed in each direction, we get the Fibonacci numbers. 5-8-13 and 8-13-21 are the typical Fibonacci sequences pineapples exhibit.

2. Arrangement of seeds on flower heads

Nature tries to optimize the arrangement of seeds on flower heads. To fill space efficiently, with no gaps from start to end, seeds must grow in a pattern that is not well approximated by a fraction. This number is exactly φ = 1.6180 with the corresponding angle (α) of 137.5 degrees. Each new seed grows at a certain angle in relation to the previous one. What’s so special about α = 137.5 degrees? It’s the golden angle! According to Dr. Mark Freitag from the University of Georgia, “if we divide a circle into two arcs in the proportion of the golden ratio, the central angle of the smaller arc marks off the golden angle, which is 137.5 degrees.” Only with this angle can one obtain the optimal filling (see fig.1 – sunflower seeds). For example, if we look at fig.2, the angle of seeds growing equals 10 degrees, and the flower in fig.3 has seeds growing 14 degrees apart.

Sunflower seeds are arranged in a beautiful spiral pattern, both in a clockwise and counter-clockwise manner. Add up the number of clockwise and counter-clockwise spirals and in 82% of the cases they will be consecutive Fibonacci numbers: 21:34, 34:55, 55:89, or 89:144. As they mature, sunflowers form more spirals.

The disk flowers perfectly arranged in the center of the head of a purple coneflower follow the same pattern as the sunflower seeds.

1. Vegetables

“Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show,”

said Bertrand Russell, and he couldn’t be more right.

Number one goes to the ultimate fractal pattern! Fractals are the result of an iterative process. They contain self-similar arrangements with different magnification factors. If you break down a fractal pattern into pieces, you get a small-size copy of the whole.

Not all vegetables are connected to the Fibonacci numbers and spirals, but those that are, are quite fascinating. Let’s take, for example, Romanesque broccoli (Roman cauliflower), one of the most beautiful vegetables I’ve ever seen! The peaked florets are arranged in beautiful Fibonacci spirals, 8 and 13 in this case.

Chinese leaves and lettuce are other vegetables that display Fibonacci phyllotaxis.

If you are wondering why I didn’t mention the Fibonacci patterns and golden ratio (allegedly) present in the structure of the Pantheon, in Da Vinci’s & Raphael’s paintings or in sea shells, it’s because much of what is presented about it in zoology, visual arts, architecture etc. is often false or seriously misleading.

“Unfortunately, these statements about the golden ratio have achieved the status of common knowledge and are widely repeated. Even current high school geometry textbooks . . . make many incorrect statements about the golden ratio. It would take a large book to document all the misinformation about the golden ratio, much of which is simply repetition of the same errors by different authors,” wrote Prof. Clement Falbo in The Golden Ratio – A Contrary Viewpoint. “The nautilus is definitely not in the shape of the golden ratio,” he continues. “Anyone with access to such a shell can see immediately that the ratio is somewhere around 4 to 3. In 1999, I measured shells of Nautilus pompilius, the chambered nautilus, in the collection at the California Academy of Sciences in San Francisco. The measurements were taken to the nearest millimeter, which gives them error bars of ±1 mm. The ratios ranged from 1.24 to 1.43, and the average was 1.33, not φ (which is approximately 1.618). Using Markowsky’s ±2% allowance for φ to be as small as 1.59, we see that 1.33 is quite far from this expanded value of φ. It seems highly unlikely that there exists any nautilus shell that is within 2% of the golden ratio.”

Read Falbo’s article or Misconceptions about the Golden Ratio by George Markowsky and decide for yourself what’s credible.